Constructions and Applications of W-States A Major Qualifying Project Submitted to the Faculty of Worcester Polytechnic Institute in partial fulfillment of the requirements for the Degree in Bachelor of Science in Physics and Mathematical Sciences By James McClung Date May 17, 2020 Advisors Professor P. K. Aravind Professor Umberto Mosco This report represents work of WPI undergraduate students submitted to the faculty as evidence of a degree requirement. WPI routinely publishes these reports on its web site without editorial or peer review. For more information about the projects program at WPI, see http://www.wpi.edu/Academics/Projects. Abstract After introducing the fundamentals of quantum circuits for a general audience, this report evaluates two means of constructing W-states: the probabilistic naive construction (PNC) and the optimized splitter construction (OSC). These methods are compared in terms of asymptotic gate cost and other parameters. Furthermore, a proof of Bell's theorem is explored with new results concerning the probability of contradicting elements of reality using W-states. i Acknowledgements Thanks to Connor Anderson for discovering the connection between (96) and Eulerian numbers. Thanks also to Ethan Washock for helping with the same. Thanks to all my friends in the math lounge, especially Kyle Dituro, for making my time there brighter, and to Mr. Greg Aubin for motivating me on late nights spent working. Thanks to my parents and to Emily Staknis for their constant support. Most of all I thank Professor P. K. Aravind for putting me to work on Qudoku over a year ago, and guiding my studies in quantum information theory all the way up to this point. ii Contents Abstract i Acknowledgements ii 1 Introduction 1 2 Quantum Circuit Preliminaries2 2.1 What Is a Qubit?...........................................2 2.1.1 The Bloch Sphere......................................3 2.1.2 Matrix Representation....................................4 2.1.3 Multiple Qubits and Entanglement.............................4 2.2 Measurement and The Born Rule..................................5 2.3 Special Quantum States.......................................6 2.3.1 Bell States..........................................6 2.3.2 GHZ States..........................................7 2.3.3 W-states............................................7 2.4 Quantum Gates............................................7 2.4.1 Single Qubit Gates......................................8 2.4.2 Multi-Qubit Gates......................................9 2.4.3 Properties of Gates...................................... 12 2.4.4 Universality of Gates and Arbitrary Approximation................... 14 2.5 Quantum Circuits.......................................... 15 2.5.1 How to Read a Circuit Diagram.............................. 15 2.5.2 Constructions of Arbitrary Controlled Gates....................... 16 3 Constructions of W-states 18 3.1 On 2k Qubits............................................. 18 3.2 On n Qubits, Sometimes....................................... 19 3.2.1 Optimized Probabilistic Naive Construction........................ 21 3.3 On n Qubits, Always but Inefficiently............................... 21 3.4 On n Qubits, Always and Efficiently................................ 23 4 Proof of Bell's Theorem using W-states, Without Inequalities 25 4.1 Elements of Reality.......................................... 25 iii 4.2 W-states in X-Basis......................................... 25 4.3 Proof Using Three Qubits...................................... 26 4.4 Proof Using More Qubits...................................... 27 5 W-states and LOCC 29 5.1 What Is LOCC?........................................... 29 5.2 LOCC Classes............................................. 29 6 Conclusion 31 References 32 Appendix A Expected Cost of Optimized Probabilistic Naive Construction 35 iv List of Figures 1 Construction of jΨ+i, the simplest W-state............................. 15 + 2 Construction of jΨ iusing C1X0................................... 15 3 Circuit identity for (62)........................................ 15 1 4 Circuit that outputs j0i in the lower qubit with probability 2 and j1i otherwise......... 16 5 Construction of arbitrary CU gate.................................. 16 6 Construction of arbitrary CCU gate, where V2 = U.......................... 17 7 Construction of reverse-controlled gates. The dotted box indicates where two X gates canceled each other out............................................. 17 8 Construction of jW4i using naive method. Left to right, the enclosed subcircuits split j0000i into 4 equally likely states; map j0000i to j0100i; and map j0011i to j1000i........... 18 9 Construction of jW8i using naive method. First block: splitting j08i into 8-state superposition. Second block: placing 1s in desired locations. Third block: removing unwanted 1s....... 19 10 Construction of jW5i in bottom five qubits via PNC. If any of the measured qubits return 1, the register is reset to 0s and the circuit is run again........................ 20 11 Construction of jW5i via OPNC. Block 1: splitting initial state into superposition of 8 states. Block 2: filtering out three extraneous states from superposition. Block 3: mapping remaining states to one-hot states........................................ 21 12 Sub-optimal construction of jW5i using splitter gates....................... 22 13 Optimal construction of jW5i using splitter gates.......................... 23 v List of Tables 1 Some notation used in this report..................................2 2 Basis states along x-, y-, and z-axes expressed as non-normalized superpositions of each other.4 3 Properties of tensor product.....................................5 4 Summary of common single-qubit gates...............................8 5 Input and output table of controlled-Z gate with different configurations of control and target. 12 6 Comparison of construction methods. \Standard gates" are those gates introduced in §2.4.1 and the CX gate............................................ 24 7 Representative states for each of the six LOCC classes on three qubits.............. 29 vi 1 Introduction Quantum computing currently receives a lot of attention due to its ability to dramatically outperform ordinary (classical) computing at many tasks. Quantum cryptography, in particular, is a still-developing field with major ramifications for cybersecurity. Peter Shor's seminal algorithm [1] makes use of quantum phenomena to factor products of two large primes, which is virtually impossible for classical computers and forms the basis of modern encryption. Quantum computers harness the power of superposition to outperform classical computers. A set of classical bits, represented by a string of 0s and 1s, represents a definite state and can be read and written to at will. By contrast, a set of quantum bits, or qubits, can be in a superposition of several bit strings at the same time. Although it is impossible to determine the overall state of a qubit, since a direct measurement \collapses" the superposition into one of its component states, it is nevertheless possible to linearly transform the entire superposition without causing collapse. These linear operations, which must be reversible, enable parallel processing of bit strings. Since adding a qubit to a register of qubits doubles the number of bit strings that can be in a superposition, the power of quantum computers grows exponentially with the number of qubits. W-states are one of two main classes of multipartite entangled states, the other being GHZ states.1 A W-state is an equally-weighted superposition of \one-hot" states, i.e. states in which a single qubit is in the state 1 and all the others are 0. For example, the 3-qubit W-state is 1 1 1 jW3i = p j001i + p j010i + p j100i : (1) 3 3 3 W-states have many applications. Jha et al. employ W-states to solve the n-queens problem [3]. W-states are also used in quantum communication [4] and cryptography [5] protocols. More generally, W-states are more robust than GHZ states in some ways [6,7] and their form can be exploited to disprove locality [8]. W-states are difficult to construct with elementary gates, however. Most constructions are complicated and tend to be specific to the underlying quantum computer [9{12]. In §3 of this report I analyze two approaches to constructing W-states, neither of which utilizes advanced gates common to other constructions. The two approaches are compared in terms of asymptotic gate cost and other parameters. A generalization of Cabello's proof of Bell's theorem using W-states [8] is given in §4. Finally, a brief explanation of LOCC classes is given in §5. Before presenting the new results obtained in this study, we first provide an introduction to the basics of quantum information theory, including quantum circuits, for a reader who is new to this field. An understanding of the material in §2 should be sufficient for an understanding of the rest this report. For a more thorough introduction to quantum information and circuits, the reader is referred to the excellent works of Nielsen and Chuang [13] and Mermin [14]. 1For more than 3 qubits, other classes include the cluster states, which have their own interesting properties [2]. 1 2 Quantum Circuit Preliminaries This section introduces some of the basic notions of quantum computer science. A reader who is already proficient in this subject may skip to §3. This section also covers the notation that I use throughout the report, and thus may